Discussion
The velocity field and the surface normal vector are given by, \(\vec{V} = u\hat{i} + v\hat{j} + w\hat{k}\) and \(\vec{n} = n_{1}\hat{i} + n_{2}\hat{j} + n_{3}\hat{k}\), respectively. If Euler equations are to be solved, the boundary condition that must be satisfied at the wall is,
Answer: A
The wall condition is \(\vec{V} \cdot \vec{n} = 0\).
The Euler equations describe inviscid flow. Without viscosity the fluid can slip along a solid wall, so the only physical requirement is that it does not pass through it: the velocity component normal to the wall must vanish, \(\vec{V} \cdot \vec{n} = 0\) (the no-penetration or slip condition).
\(\vec{V} = 0\) is the no-slip condition, which belongs to the viscous Navier-Stokes equations and cannot be imposed on the lower-order Euler equations. \(\nabla \cdot \vec{V} = 0\) is the incompressibility condition throughout the field, not a boundary condition. \(\vec{V} \times \vec{n} = 0\) would force the flow to be perpendicular to the wall, the opposite of what an inviscid wall allows.
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